Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#7162 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #102 |
| commutator | C8 | C4×C8×C19 |
| Frattini, cosocle | C2×C4×C8 | C22×C19 |
| socle | C2×C19 | #1277 |
| prime radical | C23×C19 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 2 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,30,26,32,41,49,27,33,42,31,25,43,24,29)(22,34,36,47,39,45,51,50,40,48,35,28,38,44,37,46);(20,22,24,37,25,38,42,35,27,40,41,51,26,39,23,36)(21,28,31,45,32,46,29,48,33,47,30,44,43,50,49,34)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 150);